In order to improve the control performance of the three-phase permanent magnet synchronous motor(PMSM) system, a novel sensorless adaptive sliding mode finite control set model predictive control(FCS-MPC) strategy is proposed based on the speed extended state observer(ESO) of the arc-hyperbolic sine function. The ESO is used to construct the speed and the back electromotive force(EMF) observer of the PMSM system to realize the estimation of the motor's speed and back EMF fast and accurately. The adaptive sliding mode control with the load ESO is used as the speed regulator of the system to improve the robustness of the system. The FCS-MPC strategy based on fast vector selection is used to reduce the torque ripple and reduce the computational complexity of the system algorithm. The simulation results show that the PMSM system can run reliably and stably with the sensorless adaptive sliding mode FCS-MPC strategy based on the ESO, and achieve satisfactory control effects of torque and speed. Compared with the adaptive sliding mode FCS-MPC strategy based on the integral sliding mode, the proposed strategy enables the PMSMs to possess better command-following characteristics and stronger robustness in the presence of various of reference speed and load torque.